Solid Mechanics II $($MECH$326)$ Final Exam $($Fall$,2021)$

Time allowed for $120$ minutes and attempt four questions.

A propeller shaft subjected to combined torsional and axial loads is designed as shown in Fig. P$1$a$.$ A $60-$degree strain rosette as shown in Fig P$1$b was used to obtain strains from a$,$ b and c strain gauges on the surface of the shaft. $[30$ pts$]$

$($a$)$ Derive strain components on the surface of the shaft, i$.$e$.,{}_x,{}_y$ and ${}_{xy},$ from strains obtained by the strain rosette shown in Fig. P$1$b$.[10$ pts$]$

$($b$)$ Derive the representation of the shear modulus $(G)$ using the Young's modulus $(E)$ and the Poisson's ratio $(v)$ by assuming pure shear conditions. $[10$ pts$]$

$($c$)$ Due to the loading, the strain gauges of the $60$ degree strain rosette give a reading of ${}_a=-55({10}^{-6}),{}_b=130({10}^{-6}),{}_c=75({10}^{-6}).$ Determine stress components on the surface of the shaft, i$.$e$.,{}_x,{}_y$ and ${}_{xy},$ and determine principal stresses based on strain Mohr's circle. Assume that Young's modulus and Poisson's ratio of the steel are $205$ GPa and $0\mathrm{.}35$ respectively. $[10$ pts$]$

Hint P$1.$ The strain transformation equations

${}_{x^{\text{'}}}={}_{x}co{s}^2+{}_{y}si{n}^2+{}_{xy}sincos$

${}_{x^{\text{'}}{y}^{\text{'}}}=-({}_x-{}_y)sincos+{}_{xy}(co{s}^2-si{n}^2)$